The core-triviality conjecture for finite-dimensional cocommutative cosemisimple Yetter–Drinfel'd Hopf algebras

Let $K$ be an algebraically closed field of characteristic zero, let $G$ be a finite abelian group, and let

$$H=K[G].$$

Let $A$ be a finite-dimensional cocommutative cosemisimple Yetter–Drinfel'd Hopf algebra over $H$, and let $\eta\in A$ be group-like. Denote by $G_\eta$ the index group of $\eta$. Core-triviality conjecture. The core of $\eta$ is trivial as a Yetter–Drinfel'd Hopf algebra over the group ring $K[G_\eta]$. The conjecture is motivated by the examples considered in the paper, where every core was trivial, but those examples provide only limited evidence for the general claim.